Construct Functions

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Answer Key & Explanations

Construct Functions · Grade 8 · Worksheet 1

1. Liam is tracking his savings for a new video game system. He starts with $120 in his savings account and adds $15 each week from his allowance. Write an equation in slope-intercept form that represents the total amount of money, y, Liam has after x weeks. Answer: y = 15x + 120 Solution: Liam starts with $120 in his savings account. This is the initial amount before any weeks pass. He adds $15 each week from his allowance.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Liam starts with $120 in his savings account. This is the **initial amount** before any weeks pass. He adds $15 each week from his allowance. We want an equation for the total money \( y \) after \( x \) weeks. --- **Step 2: Identify the slope-intercept form** The slope-intercept form of a linear equation is: \[ y = mx + b \] where: - \( m \) = slope (rate of change per week) - \( b \) = y-intercept (starting amount at week 0) - \( x \) = number of weeks - \( y \) = total money --- **Step 3: Determine \( b \) (y-intercept)** At \( x = 0 \) weeks, Liam has $120. So \( b = 120 \). --- **Step 4: Determine \( m \) (slope)** Each week, he adds $15. So the rate of change (slope) \( m = 15 \). --- **Step 5: Write the equation** Substitute \( m = 15 \) and \( b = 120 \) into \( y = mx + b \): \[ y = 15x + 120 \] --- **Step 6: Interpret the equation** - After 0 weeks: \( y = 15(0) + 120 = 120 \) - After 1 week: \( y = 15(1) + 120 = 135 \) - After 2 weeks: \( y = 15(2) + 120 = 150 \) This matches the situation described. --- **Final Answer:** y = 15x + 120

2. A local coffee shop is tracking the relationship between the outdoor temperature and their iced coffee sales. They found that when the temperature is 68°F, they sell 45 iced coffees, and when the temperature is 86°F, they sell 75 iced coffees. Assuming this relationship is linear, write an equation in slope-intercept form that represents the number of iced coffees sold (y) based on the temperature in °F (x). Answer: y = (5/3)x - (265/3) Solution: Linear relationships show a constant rate of change between two variables. The slope represents how much the dependent variable changes for each unit change in the independent variable.
   Full step-by-step solution

   Linear relationships show a constant rate of change between two variables. The slope represents how much the dependent variable changes for each unit change in the independent variable. The y-intercept represents the value of the dependent variable when the independent variable is zero, though this might not always have practical meaning in real-world contexts.

3. Mere earns a base salary of $40 per day plus $8 for each item she sells. Write a linear function f(x) that represents her total daily earnings when she sells x items. Answer: f(x) = 8x + 40 Solution: Identify the base amount (y-intercept, b). Mere earns $40 per day even if she sells 0 items, so b = 40. Identify the rate of change (slope, m).
   Full step-by-step solution

   Step 1: Identify the base amount (y-intercept, b). Mere earns $40 per day even if she sells 0 items, so b = 40. Step 2: Identify the rate of change (slope, m). She earns $8 for each item sold, so m = 8. Step 3: Write the linear function in the form f(x) = mx + b: f(x) = 8x + 40. The answer is f(x) = 8x + 40.

4. A local bakery is testing a new cookie recipe and needs to determine the baking time. They found that for 12 cookies, the ideal baking time is 8 minutes, and for 36 cookies, the ideal baking time is 14 minutes. Assuming the relationship between the number of cookies (x) and the baking time in minutes (y) is linear, write the equation in slope-intercept form that models this situation. Answer: y = (1/4)x + 5 Solution: Identify the two points: (12, 8) and (36, 14) Calculate the slope: m = (14 - 8) / (36 - 12) = 6 / 24 = 1/4 Use point-slope form with (12, 8): y - 8 = (1/4)(x - 12) Distribute: y - 8 = (1/4)x - 3 Add 8 to both sides: y = (1/4)x + 5 The equation is y = (1/4)x + 5.
   Full step-by-step solution

   Step 1: Identify the two points: (12, 8) and (36, 14) Step 2: Calculate the slope: m = (14 - 8) / (36 - 12) = 6 / 24 = 1/4 Step 3: Use point-slope form with (12, 8): y - 8 = (1/4)(x - 12) Step 4: Distribute: y - 8 = (1/4)x - 3 Step 5: Add 8 to both sides: y = (1/4)x + 5 The equation is y = (1/4)x + 5.

5. A taxi company charges a flat fee of $6 plus $1 per mile. Write a linear function f(x) to represent the total cost for a ride of x miles. Answer: f(x) = x + 6 Solution: Identify the flat fee (y-intercept, b) which is $6. Identify the cost per mile (slope, m) which is $1 per mile. Write the linear function in slope-intercept form: f(x) = mx + b.
   Full step-by-step solution

   Step 1: Identify the flat fee (y-intercept, b) which is $6. Step 2: Identify the cost per mile (slope, m) which is $1 per mile. Step 3: Write the linear function in slope-intercept form: f(x) = mx + b. Step 4: Substitute m = 1 and b = 6: f(x) = 1x + 6. Step 5: Simplify: f(x) = x + 6. The answer is f(x) = x + 6.

6. Aisha is comparing two internet plans. Plan A has a $30 monthly fee plus $0.50 per gigabyte of data used. Plan B has a $20 monthly fee plus $0.75 per gigabyte of data used. After how many gigabytes of data usage will both plans cost the same amount? Answer: 40 Solution: Write equations for both plans. Let x = gigabytes of data used.
   Full step-by-step solution

   Step 1: Write equations for both plans. Let x = gigabytes of data used. Plan A: Cost = 30 + 0.50x Plan B: Cost = 20 + 0.75x Step 2: Set the equations equal to find when costs are the same: 30 + 0.50x = 20 + 0.75x Step 3: Subtract 20 from both sides: 10 + 0.50x = 0.75x Step 4: Subtract 0.50x from both sides: 10 = 0.25x Step 5: Divide both sides by 0.25: x = 10 ÷ 0.25 x = 40 The answer is 40 gigabytes.

7. Maya is designing a custom bookshelf where the number of shelves (y) depends on the total height of the bookshelf in inches (x). She knows that a 48-inch tall bookshelf can have 5 shelves, and a 72-inch tall bookshelf can have 8 shelves. If the relationship between height and number of shelves is linear, how many shelves can Maya include in a bookshelf that is 96 inches tall? Answer: 11 Solution: Identify the two points from the problem: (48, 5) and (72, 8), where x is height in inches and y is number of shelves. Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1).
   Full step-by-step solution

   Step 1: Identify the two points from the problem: (48, 5) and (72, 8), where x is height in inches and y is number of shelves. Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (8 - 5) / (72 - 48) = 3 / 24 = 1/8. Step 3: Use the point-slope form to find the equation. Using point (48, 5): y - 5 = (1/8)(x - 48). Step 4: Simplify to slope-intercept form: y - 5 = (1/8)x - 6, so y = (1/8)x - 1. Step 5: Substitute x = 96 into the equation: y = (1/8)(96) - 1 = 12 - 1 = 11. The answer is 11.